D-Calib: Calibration Software for Multiple Cameras System

Transcription

1 D-Calib: Calibration Software for Multiple Cameras Sstem uko Uematsu Tomoaki Teshima Hideo Saito Keio Universit okohama Japan {u-ko tomoaki Cao Honghua Librar Inc. Japan Abstract This paper presents calibration software D-Calib for multiple cameras which can calibrate all cameras at the same time with eas operation. Our calibration method consists of two steps: initialization of camera parameters using planar marker pattern optimization of initial parameters b tracking two markers attached to a wand. B weak calibration and multi-stereo reconstruction b multiple cameras the initial parameters can be optimized in spite of few markers. Even if the cameras are placed in a large space the calibration can be achieved b swinging the wand around the space. This method is completed as commercial software which achieves 0.7 % of average errors against the marker pattern size. Introduction Estimation process of camera s unique parameters (such as focal length and skew parameters is called as camera calibration [4]. This calibration process is ver important issue for 3D reconstruction 3D motion capture and virtualized environment etc. [3 5]. DLT (Direct Linear Transformation method is commonl used as the tpical calibration method []. In DLT method multiple markers whose 3D coordinates are alread known are captured with a camera. Using the pairs of D coordinates in the captured images and known 3D coordinates of the markers twelve camera parameters (DOF is eleven are computed b least-square-method. For using DLT method in large space it is necessar to place the markers to cover the overall space. Increasing the number of markers is also necessar to improve the accurac of the calibration. However setting up the markers whose 3D coordinates is known in the large space takes amount of time and effort. Using a lot of cameras there is also a big problem to place the markers so that all of them can be captured in field of view of the camera and the do not occlude each other. Moreover it often happens that a user has to click the displa to assign D coordinates of markers. Therefore it involves a lot of effort and time when increasing the number of markers and cameras. hang proposed a eas calibration sstem based on a moving plane where checker pattern is drawn [8]. This calibration method does not need 3D calibration objects with known 3D coordinates. However since the calibration plane is not visible in all cameras the user has to move the plane in front of each camera. Moreover the sstem can calibrate onl part of the space. Therefore this sstem is not suite for using multiple cameras. Svoboda et al. proposed a calibration method for multiple cameras using one bright point like a laser pointer [7]. The point in the captured images are detected and verified b a robust RANSAC. The structures of the projected points are reconstructed b factorization. These computations can be automaticall done. Since one point is used as a calibration object however onl re-projection errors are evaluated in their method. In this paper we introduce calibration software D- Calib designed for multiple cameras using in a large space []. In this software we can estimate the camera parameters of all cameras at the same time b using a square pattern with si spheral markers and a wand with two spheral markers. After capturing one frame of the square pattern the wand swinging in the whole of space is captured for a few hundreds of frames. Therefore we do not have to set up a lot of markers around the space. The spheral markers are made b retroreflective materials so that the can be successfull detected from the captured images. Since all the processes are automaticall performed b image processing technique without manual operations the user s task is etremel reduced. In our method initial parameters computed from the square pattern are optimized b using the two markers attached to the wand. Since the distance between the markers is known we can also optimize 3D reconstruction errors in addition to D re-projection errors which are onl accurac evaluation in Svoboda s method. 4th International Conference on Image Analsis and Processing (ICIAP /07 $

2 Input images Cam Cam Cam n Binarization & Labeling Detect markers Input images Cam Cam Cam n Binarization & Labeling Track & Identif Markers Calc homograph Calc D errors Calc 3D errors Estimate init params. Optimization Estimation of initial parameters p p p 34 p' p' p' 34 Optimization of initial parameters Figure. Flowchart of calibration process. Camera Calibration The relationship between the 3D real world ( and the D image ( is represented as a following equation. P p p p 3 p 4 p p p 3 p 4 p 3 p 3 p 33 p 34 ( This 3 4 matri P is called as a projection matri and the twelve elements of the matri are called as camera parameters. Estimation of these camera parameters corresponds to camera calibration. In our software the twelve camera parameters are computed for each camera. Since we focus on computing accurate etrinsic parameters of multiple cameras we assume that intrinsic parameters of the cameras are previousl given b using other method. If using uncalibrated cameras the intrinsic parameters are approimated b simple form without skew and distortion parameters as described in Sec. A.. 3 Our Calibration Method Fig. shows the flowchart of the calibration method in our software. This method consists of two steps: ( estimation of initial parameters ( optimization of initial parameters. In the estimation of initial parameters a square pattern with si spheral markers is placed on the floor and is captured b multiple cameras as shown in Fig. (a. The real positions of the spheral markers on the square pattern are previousl known. Using the relationship between the D positions of the markers in captured images and the 3D positions on the markers camera parameters are computed for each camera as initial parameters. These camera parameters represent twelve elements of a projection matri P as described in Sec.. Actuall the etrinsic parameters of each camera are computed from the square pattern and combined with the intrinsic parameters obtained in advance into the camera parameters. Since the initial parameters are computed from the square planar pattern the accurac against perpendicular direction to the plane is relativel small compared with the other directions. Moreover high calibration accurac can be obtained onl in the area close to the square pattern. Therefore the initial parameters of ever camera should be improved b the optimization at the same time in the net square pattern P P P 3 (a Capturing images for estimation of initial parameters. wand P' P' P' 3 (b Capturing images for optimization of initial parameters. Figure. Sstem environments in our software. 4th International Conference on Image Analsis and Processing (ICIAP /07 $

3 step. In the optimization step we collect a hundreds of frames captured in the multiple cameras as shown in Fig. (b while a wand that has two spheral markers is swung in the entire object space. The positions of the two spheral markers are detected and tracked from the captured image sequences of ever camera. Since all the spheral markers are made b retroreflective materials the can be easil detected from the captured images. B evaluating the estimated etrinsic parameters with epipolar constraints of tracked markers and the 3D distance between the two markers on the wand the parameters are optimized. 3. Estimation of Initial Parameters A square pattern with si spheral markers is placed on the floor. Then images of the square markers are captured b multiple cameras. From each captured image the si markers D coordinates are detected as ( ( 6 6. We assume that the plane of the square pattern is - plane in the 3D coordinate. Then we define 3D coordinates of the si markers as ( 0 ( Based on this definition the relationship between the D and 3D coordinates of the markers can be represented b using a 3 3 planar transformation matri (Homograph H as following equation. ( i 6 i i H i i h h h 3 h h h 3 h 3 h 3 h 33 i i ( This equation is equivalent to eq. ( in Sec. when = 0 in eq. (. This means that H provides the first second and fourth column vectors of the projection matri P. H is a matri which transfers points between the planes and is easil computed b more than four corresponding points on both of the planes. Accordingl we compute H between the 3D plane (- plane and the D plane (- plane from si corresponding points which are the spheral markers of the square pattern. Then the twelve camera parameters of each camera Figure 3. Corresponding of planes b homograph. H are computed from H. The detail of the computing process will be described in Sec. A. 3. Optimization of Initial Parameters In this process we define an error function consisting of D re-projection errors and 3D reconstruction errors which is used for optimizing the initial parameters b minimizing the function. As shown in Fig. (b the images of the swinging wand with two spherical markers are captured for a hundreds frames b the multiple cameras. B detecting and tracking the two spheral markers from the captured image sequences we can obtain a set of D-D corresponding points between the cameras which provides epipolar relationship among the cameras. Using this data set we can compute D re-projection errors of the initial camera parameters which are evaluated b the distance between the point of the marker in the image and the re-projected epipolar line onto the other cameras. Since the 3D distance between the two markers on the wand are known we also evaluate the accurac of the initial camera parameters b comparing the 3D distance between two marker positions recovered from the input images with the real distance between them. 3.. Tracking of Markers After binarization and labeling are performed to the captured image sequence from each camera two spheral markers regions are detected from each image. For tracking the two markers each marker must be identified through the image sequence. The identification of each marker is performed b computing the distance of the positions between the previous frame and the net frame. The marker with the closest position to the position in the previous frame is identified as the same marker. Even though the two markers are occluded each other or are not detected in some frames each marker can be continuousl tracked b using direction of marker s movement. Therefore the cameras do not have to see all the markers through the captured images. Onl the frames which are captured when the markers are visible in all cameras are automaticall selected as a data set of D-D correspondences. 3.. Optimization Using the D coordinates of the tracked markers two kinds of errors are computed; (a D re-projection errors ɛ (b 3D reconstruction errors ɛ. Error function is defined as eq. (3 and minimized b Down-hill simple method to optimize the si etrinsic parameters included in the initial 4th International Conference on Image Analsis and Processing (ICIAP /07 $

4 camera parameters; rotation (3 DOF: θ φ ψ translation (3 DOF: t t t z. If all cameras have the same intrinsic parameters it is possible to optimize seven parameters including the focal length f. This is because D re-projection errors are not absolute but relative values among the cameras. { cost(θ φψ t t t z = (ɛ + ɛ cost(θ φψ t t t z f = (ɛ + ɛ D re-projection errors ɛ When a set of D-D correspondences between two cameras is known a point in the image captured b the camera is projected onto the image captured b the camera as a line and should eist anwhere on the line. This is called as epipolar constraints and the line is called as epipolar line. The marker in the image of the camera is projected onto the image of the camera as a line (epipolar line b the initial camera parameters as shown in Fig. 4(a. The epipolar line should be projected on the position of the marker which is captured b the camera however it might be projected awa from the marker because of the P P C C Cam Cam c c c epipolar line d (a Errors from epipolar constraints. Cam (base camera (3 errors included in the initial parameters. Therefore the distance between the epipolar line and the marker in the image of camera is defined as a D re-projection error from the camera to the camera. In the same wa D re-projection errors are computed mutuall among all the cameras and then the sum of them becomes ɛ. Here the D re-projection errors are computed using epipolar constraints which represent relative geometr between the cameras. Therefore we define one camera as an absolute basis and do not compute the re-projection errors to the base camera as shown in Fig. 4(b. 3D reconstruction errors ɛ When camera parameters of two cameras are known 3D coordinates of an object which is captured b the cameras can be reconstructed b using triangulation from D coordinates in the images. In our method 3D reconstruction errors ɛ are computed b multi-stereo reconstruction of multiple cameras. When the two markers are captured b each camera as shown in Fig. 5 3D coordinates of the markers can be obtained b triangulation from the D coordinates in each image (see Sec. B. Computing the 3D coordinates of the markers from all the cameras 3D distance D between the markers can be computed. Since the real distance s is known the difference between D and s becomes a 3D reconstruction error at ever frame and then the sum of them becomes ɛ. ɛ = frame (D s After computing ɛ and ɛ the initial parameters are optimized b minimizing the error function of eq. (3 b Down-hill simple method. 4 Eperimental Results In this section we will show two eperimental results of our calibration software; four cameras configuration and ( (4 Cam Cam 3 Cam 4 s D ( P P ( ( ( ( (b Relationship of projection between cameras. Figure 4. Computation of D re-projection errors. Figure 5. Computation of 3D reconstruction errors. 4th International Conference on Image Analsis and Processing (ICIAP /07 $

5 Table. Evaluation results from optimized parameters. (a Square pattern of markers (b Wand Figure 6. Markers used in our software. numbers of cameras Minimum error [mm] Maimum error [mm] Average error [mm] Standard deviation Size of [mm] calibrated [mm] space [mm] Table. Comparison of results between initial and optimized parameters. Minimum error [mm] Maimum error [mm] Average error [mm] Standard deviation Figure 7. 3D coordinate sstem defined b initial parameters. Figure 8. Tracking results of two markers of wand. si cameras configuration. In both of the eperiments the square pattern with si markers is captured at one frame b each camera to estimate initial camera parameters as shown in Fig. 6(a. Net the wand swinging in the space is captured for 00 frames b each camera as shown in Fig. 6(b. The size of the space where the wand is swinging is about m m m. In both of the eperiments the size of square pattern marker is [mm] the distance between the two markers on the wand is 50 [mm]. The resolution of the captured image is Fig. 7 shows the results of - - coordinate sstems 4th International Conference on Image Analsis and Processing (ICIAP /07 $ initial optimized defined b initial parameters estimated b four cameras. Fig. 8 shows the tracking results of two markers which are attached to the wand b the four cameras. We can see that the two markers are successfull detected and tracked through the captured image sequence without confusing the markers b our tracking method. Table shows the evaluation results of errors obtained b optimization of the initial parameters using the tracked markers. These errors represent 3D reconstruction errors which are computed using optimized camera parameters instead of the initial parameters in the same wa described in Sec The error is computed at ever frame in the captured image sequence and then minimum maimum and average values and standard deviation of the computed errors are listed in Table. Table shows each error computed b using the initial and the optimized parameters. From Tab. we can find that the minimum errors are less than 0. [mm] and the maimum error is also about 4.6 [mm]. These results can be considered small enough compared to the size of the space. The errors included in the initial parameters can be reduced b our optimization method as shown in Tab.. Moreover the standard deviation shows that the space can be impartiall calibrated b swinging the wand around the space. 5 Conclusions In this paper we proposed calibration software for multiple cameras. The camera parameters of all the cameras can

6 be estimated at the same time. Since all the processing of this software can be automaticall performed without user s manual operations like a assigning the marker s positions b clicking the displa the user can easil use this software. This software can calibrate a lot of cameras which are placed in large space without placing man markers around the space in advance. Therefore it is well suited for the task which needs large space like 3D motion capture. A Calculation P from H A projection matri consists of intrinsic and etrinsic parameters. Therefore we obtain a projection matri b computing the intrinsic and the etrinsic parameters from the homograph and combining them [6]. We define intrinsic parameters a rotation matri and a translation vector of etrinsic parameters as A R and t. Then a projection matri P will be P = A [R t] = A [r r r 3 t] (5 A = f 0 c 0 f c 0 0 (6 r r r 3 are column vectors of the rotation matri R f is the focal length (c c is a center of the image. From eq. ( P = A [r r t] = H = h h h 3 h h h 3 h 3 h 3 h 33 (7 A H = [r r t] (8 Using the homograph H computed from si corresponding points of spheral markers the intrinsic and etrinsic parameters are computed from eq. (8. When both the intrinsic and etrinsic parameters are obtained the projection matri P is obtained b (5. A. Computation of Intrinsic Parameters When using uncalibrated cameras intrinsic parameters of the cameras have to be computed. Since we define A as eq. (6 we onl have to obtain a focal length f. Using the propert of the rotation matri R that inner product of r and r is 0 f is obtained b developing eq. (8 as follows. f = (h c h 3 (h c h 3 h 3 h 3 + (h c h 3 (h c h 3 h 3 h 3 (9 A. Computation of Etrinsic Parameters The first and second column vectors r r of the rotation matri R and the translation vector t are alread obtained when H is computed from eq. (8. Therefore we onl have to compute the third column vector r 3 of R. Also using the propert of R that the cross product of r and r becomes r 3 we obtain it as follows. B R = [r r r 3 ] = [r r (r r ] (0 Multi-Stereo Reconstruction When camera parameters of multiple cameras are known a 3D coordinate of a point ( can be computed from following equation using D coordinates of the corresponding points ( i i in each camera. (p 3 p (p 3 p (p 33 p 3 (p 3 p (p 3 p (p 33 p 3. (p n 3 n p n (p n 3 n p n (p n 33 n p n 3 (p n 3 n p n (p n 3 n p n (p n 33 n p n 3 References = p 4 p 34 p 4 p 34 p n 4 p n 34 n p n 4 p n 34 n ( [] []. I. Abdel-Aziz and H. M. Karara. Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetr. In Proc. of Smposium on Close-Range Photogrammetr pages [3] G. K. M. Cheung S. Baker and T. Kanade. Shape-fromsilhouette of articulated objects and its use for human bod kinematics estimation and motion capture. In Proc. of Computer Vision and Pattern Recognition pages June 003. [4] R. Hartle and A. isserman. Multiple View Geometr in computer vision. CAMBRIDGE UNIVERSIT PRESS 000. [5] H. Saito S. Baba and T. Kanade. Appearance-based virtual view generation from multi-camera videos captured in the 3d room. IEEE Transactions on Multimedia 5(3: September 00. [6] G. Simon A. Fitzgibbon and A. isserman. Markerless tracking using planar structures in the scene. In Proc. of the ISAR pages [7] T. Svoboda D. Martinec and T. Pajdla. A convenient multicamera self-calibration for virtual environments. Presence 4(4:407 4 August 005. [8]. hang. A fleible new technique for camera calibration. IEEE Transactions on Pattern Analsis and Machine Intelligence (: th International Conference on Image Analsis and Processing (ICIAP /07 $